Crop Factor: What It Changes and What It Does Not

Crop factor is one division, and it is a division of diagonals

The reference is the 35 mm film frame, 36 × 24 mm, whose diagonal is √(36² + 24²) = √1872 = 43.2666 mm. The crop factor of any other sensor is that number divided by its own diagonal. Nikon and Sony DX measure 23.6 × 15.7 mm, so their diagonal is 28.345 mm and the factor is 43.2666 / 28.345 = 1.5264. Canon APS-C measures 22.3 × 14.9 mm, diagonal 26.820 mm, factor 1.6132 — which is why the two APS-C families are not interchangeable in an equivalence calculation. Micro Four Thirds measures 17.3 × 13.0 mm, diagonal 21.640 mm, factor 1.9994, close enough to 2 that everyone rounds it. The 1-inch type measures 13.2 × 8.8 mm, diagonal 15.864 mm, factor 2.7273 — and the name refers to a vidicon tube convention, not to any dimension of the sensor.

The lens does not change; the rectangle behind it does

A 50 mm lens projects a circular image whose geometry is fixed by the glass. The focal length is a property of that glass, measured on an optical bench, and no camera body can alter it. What a smaller sensor does is record a smaller rectangle cut out of the same projected circle. Nothing is magnified. Something is discarded. A DX sensor placed behind that 50 mm records 23.6 × 15.7 = 370.5 mm² of the image where full frame records 36 × 24 = 864 mm² — that is 42.9% of the area, and the other 57.1% simply falls outside the sensor.

The angle of view formula makes this arithmetic instead of rhetoric. For a sensor dimension d and a focal length f, the angle covered is 2·arctan(d/2f). Put the same 50 mm on both bodies and compute horizontally: full frame gives 2·arctan(36/100) = 39.60°, DX gives 2·arctan(23.6/100) = 26.56°. Diagonally, 46.79° against 31.65°. The lens term f is 50 in both calculations. Only d changed. That is the whole mechanism, and it contains no magnification anywhere.

The framing equivalence, computed both ways

Now the useful question: which lens on full frame frames exactly what the 50 mm frames on DX? Solve 2·arctan(23.6/2·50) = 2·arctan(36/2·f) for f, or take the shortcut of multiplying by the crop factor: 50 × 1.5264 = 76.32 mm. Check it. A 76.32 mm on full frame gives 2·arctan(36/152.64) = 26.54° horizontally and 31.65° diagonally, matching the DX 50 mm to two decimal places. The equivalence is exact, and it is exact because the crop factor was defined as a diagonal ratio in the first place.

This is also where the popular shorthand quietly loses accuracy. The stock phrase is that a 50 mm on APS-C behaves like a 75 mm, because 1.5 is easier to say than 1.5264. A real 75 mm on full frame gives 26.99° horizontally, which is 1.63% wider than the 26.56° the DX 50 mm actually delivers. That is small, but it is a systematic bias, and on Canon APS-C — where the true factor is 1.6132, not 1.5 — the same shorthand is out by a good deal more. If the framing has to match, use the calculator and the real sensor dimensions.

The half that gets omitted: the f-number multiplies too

Almost every crop-factor explanation stops after the focal length, and that is where the confusion about equivalent aperture begins. Recall that N = f/D, so the physical diameter of the opening is D = f/N. Take full frame 50 mm at f/4: D = 12.5 mm. Now put the equivalent 25 mm on Micro Four Thirds and keep f/4: D = 6.25 mm, half the diameter, a quarter the area. Same framing, but the lens is now gathering a quarter of the light per unit of scene and projecting it onto a sensor with a quarter of the area — the two effects cancel in brightness, which is why f/4 meters the same on every format, and they do not cancel in anything else.

To match the depth of field and the total number of photons, D must stay at 12.5 mm, which means the f-number has to be divided by the crop factor on the smaller format — equivalently, multiplied by it when converting the other way. On Micro Four Thirds, 25 mm at f/2 gives D = 25/2 = 12.5 mm. On Nikon DX, 32.8 mm at f/2.62 gives 12.5 mm. On Canon APS-C, 31.0 mm at f/2.48. On a 1-inch sensor, 18.3 mm at f/1.47. Every row of the table above holds D constant at 12.50 mm, and that single constraint is what makes the depth of field match to within about 1% across a 2.7× range of sensor sizes.

The ISO column completes the picture and is the reason the equivalence is not marketing nonsense. Because f/2 admits four times the illuminance of f/4, the Micro Four Thirds body reaches the same file brightness at the same 1/250 s with ISO 100 where the full-frame body needed ISO 400. Same framing, same depth of field, same shutter, same total photons, same shot noise, four times lower ISO number. The two systems really are doing the same thing; they are just labelling it differently. Full frame 50 mm f/4 ISO 400 and Micro Four Thirds 25 mm f/2 ISO 100 are the same photograph.

Where the crop factor genuinely stops working

It is a diagonal ratio, so it equalises the diagonal and nothing else. Full frame is 3:2 and Micro Four Thirds is 4:3, so a 25 mm on Micro Four Thirds matches a full-frame 50 mm diagonally — 46.81° against 46.79° — while covering 38.17° horizontally against 39.60°, and 29.15° vertically against 26.99°. The equivalent lens is narrower and taller. If your composition depends on the width of the frame, the crop-factor conversion is the wrong tool and you should compute the horizontal angle directly.

The same mismatch shows up in area. Where the aspect ratios agree, area scales exactly as the square of the crop factor: full frame over Nikon DX is 864 / 370.52 = 2.3319 against a crop factor squared of 2.3300. Where they disagree it does not: full frame over Micro Four Thirds is 864 / 224.9 = 3.8417 against a crop factor squared of 3.9975, so Micro Four Thirds holds 4.1% more sensor area than its crop factor implies. That is why the total-light row in the table comes out at 1.041 rather than 1.000 for that format — a real, small advantage that the diagonal-based shorthand hides.

Finally, the crop factor says nothing at all about resolution, and nothing at all about perspective. Resolution is pixel count and pixel pitch, a separate axis entirely: a 24-megapixel DX sensor has a 3.93 µm pitch where a 24-megapixel full-frame sensor has 6.00 µm. Perspective — the relative size of near and far objects, what people call compression — depends only on where the camera stands. Two bodies at the same spot with the same framing record the same perspective, whatever their sensors measure.

Frequently asked questions

Does a 50 mm become a 75 mm on an APS-C body?

No. It stays a 50 mm — the focal length is a property of the glass and is unchanged by the body behind it. What changes is coverage: 39.60° horizontally on full frame becomes 26.56° on Nikon DX, because the smaller sensor records only 42.9% of the area the larger one records from the same projected image. The 75 mm figure is a framing comparison, not a transformation, and it is slightly wrong even as a comparison: the exact match is 50 × 1.5264 = 76.32 mm, and a real 75 mm covers 26.99°, which is 1.63% wider.

Do I have to apply the crop factor to the f-number when metering?

No, and this is the distinction that makes equivalent aperture look like nonsense when it is stated carelessly. Exposure is illuminance per unit of sensor area, and that depends only on the f-number. f/4 at 1/250 s and ISO 400 gives the same file brightness on full frame, on APS-C, on Micro Four Thirds and on a phone. The crop factor applies to the f-number only when you are matching depth of field, total collected light and therefore noise — never when you are setting the meter.

Does a crop sensor give a telephoto lens extra reach?

It gives narrower framing for free, which is not the same as more detail. Whether you actually gain depends on pixel density, not on crop factor. A 24-megapixel DX file has a 3.93 µm pitch; a 24-megapixel full-frame file cropped to the same framing keeps only 3933 × 2617 pixels, which is 10.29 megapixels — a factor of 2.33, exactly the crop factor squared. So with equal megapixel counts the crop body genuinely wins on subject detail. Against a 60-megapixel full-frame body, cropping to the DX rectangle still leaves about 26 megapixels and the advantage disappears.

Is equivalent aperture just marketing?

No, it is a statement about the physical aperture diameter D = f/N. Matching D across formats is what makes depth of field and total collected light agree, and it is checkable: full frame 50 mm f/4, Nikon DX 32.8 mm f/2.62, Canon APS-C 31.0 mm f/2.48, Micro Four Thirds 25 mm f/2 and a 1-inch sensor at 18.3 mm f/1.47 all give D = 12.50 mm, and all give the same depth of field within about 1%. What is fair to criticise is the shorthand that reports an equivalent aperture without saying it applies to depth of field and noise but not to metering — that omission is what makes readers think the number is fictional.

Why do Canon and Nikon APS-C have different crop factors?

Because APS-C is a size class, not a standard. Canon builds its APS-C sensors at 22.3 × 14.9 mm, giving a diagonal of 26.820 mm and a crop factor of 1.6132. Nikon and Sony build theirs at 23.6 × 15.7 mm, a diagonal of 28.345 mm and a factor of 1.5264. The gap is 5.7%, enough to matter when you are matching two systems shot for shot. Micro Four Thirds is different in this respect: 17.3 × 13.0 mm is written into the system standard, so every body carries the same 1.9994.